Math20C Midterm1 - Yea P Math 200 Calculus Midterm Exam I...

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Unformatted text preview: Yea P Math 200 Calculus Midterm Exam I January 31, 2007 Name: _ 11300 Dec; (3.1:) i214 - Cc' moi-‘1 \ Lecture Hour: Section Hour: Guidelines for the test: 0 N0 books, notes, or calculators are allowed. a You may leave answers in symbolic form, like m, unless they simplify further, like \/§ = 3,60 = 1, or cos(3ir/4) = —\/§/2. 0 Use the space provided. If necessary, write “see other side" and continue working on the back of the same sheet. I Circle your final answers when relevant. 0 Show all steps in your solutions and make your reasoning clear. Answers with no explanation will receive no credit, even if they are correct. 0 No credit will be given for work/answers that are illegible 0 You have 50 minutes. Question Perfect Score Your Score /0 (l) (a) (i) Show that the following equation represents a Sphere. 2:1:2-4— 23:2 +222 —4x+4z = 5 2Cx1+vi+313Lx Has-- 5 zfxt-Zx +\ +y1' .;--2"'-+‘—:?.2“t"I "3—5 4 l "A" 2([ x— iii-.4. x/z- +(2-+\\)‘)= Z ‘Z. '2 (x 431+ 714414 ‘3‘”: 3; "Evhs '15. om Gdtuocflrimfi 3" 951,4qu rel/v“) tar-(mat 'Wi Wuw “Hoax—i (v—a31+(\-J'b3L+L1'C317V1 (T) an—c, Anus :3ng fawn—{1m ‘15 ‘5‘ 'i'Dl-w V‘k— (ii) What is the center and radius of the sphere ? U. gem-kw (\,o,-I‘j r: \?X (b) The point (1, 2, 3) rests on the surface of a Sphere of radius r centered at the origin. Find r. Page I (2) (b) Suppose that a and b are vectors as shown below. Sketch the vector (1 + (1, clearly showing its relation with respect to 5: and b '3"! 9l Suppose that 1'1 = (1, 1, 0) as shown below. Find and state the coor- dinates of two vectors ’5' and 13 such that 11' x 17 = 11' x 117 but 13% 11'}. :MV : (wm- Mavm,‘u_.us+ Mav.,u1V1‘ 911‘“) Z Page 2‘ (3) (a) Match the foliowing equations (i)-(iii) with the graphs (I)—(III). Y0u need not explain your decisions. $213=cost y=sint, z=sin5t @ (ii) 3 = cos 2Ut, y = 81112015, 2 = logt (11) (III) (b) Sketch the curve with the vector equation fit) = t§+ e“ :3. Indicate with an arrow the direction in which 15 increases. >< / '2 I '2’ 'f I: 7‘ 2” L ‘ :3 q l II-l | :é - 1 " 2 6”” / \ 3. 3 .6'3 Page 3 (4) A particle moves along a. curve C with velocity at time t given by "3(t) = (4cos t) 3+ 3}" -— (4 sin 15) E. Suppose that at time t = 0, the particle is at location given by the vector 13:41:. (3.) Find the vector functiori F(t) which gives the position of the particle at any time t 2 0. Sleerl g Steam-l: +13” "(Lian/1W2 = can it}? +3+A§+Hcowfr¢ ...-—'I {fir-‘1‘" em; crow/7: + big «(”03“ +26% “7.. .V Reparametrize C with respect to arc length 5 measured from the ‘ point where t = 0 in the direction of increasing t. 3H): 3; lF‘Cm ldu (c) How far is this particle from its initial position when time t = 7: ? a a “were! , a (3%)-(3):(3%) rL-hlii-ls'inflxi'Biif/«tiblcos 7r+qlic 0 <4 44 M“ 01" ”Tr/35+ fawn: W vCfl=OA4EW§+OE cwm .: Sf _ ) Find the tangential (gorfiponent of the acceleration vector when t = ar. 21‘ - v‘ “if + k: :3 W-i- an r, C'L'I't-i‘xr'K v':1v'Hll:-% [Limit/1 + 5,: - (Lia‘millé v l~43m+f+ a; e 4co5+ 12/ Page 4 v': l'LiS'lfl‘HC # qus-tgl V‘ ; ”(ilel'l'x-t +CO§+TZI> (5) Suppose the planes P, Q are given by P : :1: + y — z = 2; Q : 2m—y+32=1. (a) Let L denote the line of intersection of P and Q. Find a. point on L. ‘. '2“ 2:0 fl.VW1:(”1I)(-é)g<3'i:‘(3413,“l'QD swig:);,,z Yz2’)‘ (ire we? ZX-V“ Yn‘V' X_{_\/.-i_2 _ 'I 2,>(:QX"' 2 F'3 "'3 ('0) 3=32< ' xii Y: (b) Find a vector 11‘ which is parallel to L. (c) Find a. plane which is perpendicular to L and contains the point (1,0,1). —‘éb><+8 +\/ +72 -7=CJ '9X+\/ +71: ~ _ PageE...
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